Absence of squirt singularities for the multi-phase Muskat problem

arXiv:0911.4109v1 [math.AP] 20 Nov 2009 Absence of squirt singularities for the multi-phase Muskat problem Diego C´ordoba and Francisco Gancedo November 20, 2009 Abstract In this paper we study the evolution of multiple fluids with different constant densities in porous media. This physical scenario is known as the Muskat and the (multi-phase) Hele-Shaw problems. In this context we prove that the fluids do not develop squirt singularities. 1 Introduction We consider the dynamics of the two interphases between three incompressible and immiscible fluids in a porous medium without surface tension. The free boundaries are prescribed by the jump of densities between the fluids. Although the present paper is devoted to the evolution of two interphases in R3, the same approach can be performed when additional phases are considered in R2or R3. The governing equations for the 2D incompressible porous media are identical with those modeling the dynamics in a Hele-Shaw cell (see [17]). The precise formulation of this problem is as follows [1]: the scalar density ρ=ρ(x, t) of the fluid is convected by the incompressible velocity flow u= (u1(x, t), u2(x, t), u3(x, t)) which satisfies Darcy’s law i.e. ρt+u·∇ρ= 0 (Conservation of mass) ∇·u= 0 (Incompressibility) u=−∇p−(0,0, ρ) (Darcy’s law) (1) where the scalar p=p(x, t) is the pressure and the acceleration due to gravity is taken equal to one to simplify the notation. Darcy’s law yields the velocity in terms of the density by means of singular integral operators as follows: u(x, t) = PV ZR3 K(x−y)ρ(y, t)dy −2 3(0,0, ρ(x)) , x ∈R3,(2) where the kernel Kis given by K(x) = 1 4π3x1x3 |x|5,3x2x3 |x|5,2x2 3−x2 1−x2 2 |x|5. 1 The integral operator is defined in the Fourier side by bu(ξ) = (ξ1ξ3 |ξ|2,ξ2ξ3 |ξ|2,−ξ2 1+ξ2 2 |ξ|2)bρ(ξ) that shows that the velocity and the density are at the same level in terms of regularity. The fluid is characterized by three different constant values of density ρ1< ρ2< ρ3 ρ(x1, x2, x3, t) =    ρ1in Ω1={x3> f(x1, x2, t)}, ρ2in Ω2={f(x1, x2, t)> x3> g(x1, x2, t)}, ρ3in Ω3={g(x1, x2, t)> x3}, where f(x1, x2, t)> g(x1, x2, t). The moving surfaces Sf(x1, x2, t) = {(x1, x2, x3)∈R3:x3=f(x1, x2, t)} Sg(x1, x2, t) = {(x1, x2, x3)∈R3:x3=g(x1, x2, t)} have the property (see below in section 2) that they can be parameterized as a graph for all time. Since the flow is incompressible the velocity of each interphase is continuous in the normal direction to the moving surface. Moreover, from the formulation it implies that the pressures are equal across the surfaces. The system is highly non-local in the sense that the equation for the moving surfaces involves singular integral operators and they are coupled together. Within the formulation we can recover the dynamics of a single interphase by taking ρ1=ρ2or ρ2=ρ3, which has been shown to be well-possed in the stable scenario with a maximum principle (see [7] and [8]). This case is known as the Muskat problem [16] (in 2D is also known as the two-phase Hele-Shaw) which has been broadly studied in [4], [11], [10], [18], [7], [8] and reference therein. The aim of this paper is first to show that for multiple interphases, in the stable case (ρ1< ρ2< ρ3), the system is well possed in a chain of Sobolev spaces. In the unstable case (ρ1> ρ2or ρ2> ρ3) there is Rayleigh-Taylor instability [7] and the system is ill-possed. Secondly we rule out a squirt singularity in the three phase system, i.e. that both interphases can not collapse in such a way that a positive volume of fluid between the interphases it gets ejected in finite time. Lets assume that both surfaces collapse at time Tin a domain Dsuch that lim t→T−[f(x1, x2, t)−g(x1, x2, t)] = 0 for (x1−ex1)2+ (x2−ex2)2< a2 where the constant a > 0 and the point ex= (ex1,ex2,ex3)∈Dare fixed. Consider the domain Ω(t) denoted by Ω(t) = {(x1, x2, x3) : (x1−ex1)2+ (x2−ex2)2≤(R(t))2, g(x1, x2, t)≤x3≤f(x1, x2, t)} with 0 < R(t)< a and a section of its boundary given by S(t) = {(x1, x2, x3) : (x1−ex1)2+ (x2−ex2)2= (R(t))2, g(x1, x2, t)≤x3≤f(x1, x2, t)}. Then by using the divergence free vector field uit follows that d dt Vol Ω(t) = ZS(t) [R′(t)−u·ν]d(Area) 2 where νis the unit normal to S(t). If the integral RT 0|u|L∞dt is bounded then we can choose a time t0∈[0, T) and take R(t) = 1 2a−ZT t|u|L∞dτ for t0≤t < T such that 0 < R(t)< a for all t∈[t0, T). Consequently d dt Vol Ω(t)≥0 for all t∈[t0, T) which prevents a collapse (squirt singularity) forming in between the moving surfaces (for more details see [5]). For a general n-dimensional definition of a squirt singularity see [6]. Therefore, the third part of the paper is devoted to show a bound of the velocity of the fluid in terms of C1,γ norms (0 < γ < 1) of the free boundary. The estimate is based on the property that in the principal value (2) the mean of the kernels Kare zero on hemispheres. This extra cancellation was used by Bertozzi and Constantin [2] for the vortex patch problem of the 2D Euler equation to prove no formation of singularities. For this system the convected vorticity takes constant values in disjoint domains and is related with the incompressible velocity by the Biot-Savart law (see [14] for more details). Let us point out that the system we are dealing with is a more singular one. Also, we quote the work [15] of Mateu, Orobitg and Verdera where this cancelation was used in quasiconformal mappings theory. Finally we would like to emphasize how the character of the kernels becomes crucial here since for analogous active scalar models we can not obtain this result. For the 2D surface quasi-geostrophic equation (SQG) [3] the kernels are odd (Riesz transforms) and for the patch problem [12] the velocity is not in L∞, it is in BMO (see [19] to get the definition and properties of the BMO space). Furthermore in the case of regular initial data for SQG and the system (1) the problem is also open [9]. The structure of the article is as follows. In section 2 we derived the equations of both interphases that are coupled together and in section 3 we show that the system is well-possed in Sobolev spaces. Finally, in section 4, we give a proof of boundedness of the velocity in terms of the smoothness of the interphases. 2 The evolution equation of the moving surfaces The goal is to obtain the dynamics of a fluid that takes three different constant values of density ρ1, ρ2and ρ3as follows ρ(x1, x2, x3, t) =    ρ1in {x3> f(x, t)}, ρ2in {f(x, t)> x3> g(x, t)}, ρ3in {g(x, t)> x3}, (3) with f(x, t)> g(x, t) for all x= (x1, x2)∈R2. Hence ∇ρ= (ρ2−ρ1)(∂x1f, ∂x2f, −1)δ(x3−f(x, t)) + (ρ3−ρ2)(∂x1g, ∂x2g, −1)δ(x3−g(x, t)), where δis the Dirac distribution defined by < hδ, η >=ZR2 h(x)η(x, f(x, t))dx, 3 for η(x1, x2, x3) a test function. For a divergence free velocity field, Darcy’s law provides u= (∂x1∆−1∂x3ρ, ∂x2∆−1∂x3ρ, −∂x1∆−1∂x1ρ−∂x2∆−1∂x2ρ),(4) and therefore u(x1, x2, x3, t) = −ρ2−ρ1 4πPV ZR2 (y1, y2,∇f(x−y, t)·y) [|y|2+ (x3−f(x−y, t))2]3/2dy −ρ2−ρ1 4πPV ZR2 (y1, y2,∇g(x−y, t)·y) [|y|2+ (x3−g(x−y, t))2]3/2dy, (5) for f(x, t)6=x36=g(x, t). Taking x3→f(x, t) yields u(x, f(x, t), t) = −ρ2−ρ1 4πPV ZR2 (y1, y2,∇f(x−y, t)·y) [|y|2+ (f(x, t)−f(x−y, t))2]3/2dy −ρ3−ρ2 4πPV ZR2 (y1, y2,∇g(x−y, t)·y) [|y|2+ (f(x, t)−g(x−y, t))2]3/2dy, without considering tangential terms. In any case the evolution of the surfaces are given by the normal velocity, the tangential terms are related with the parameterization of the free boundary [13]. Consequently (x, f(x, t))t·(−∂x1f, −∂x2f, 1) = u(x, f(x, t), t)·(−∂x1f, −∂x2f, 1), and therefore ft(x, t) =ρ2−ρ1 4πPV ZR2 (∇f(x, t)−∇f(x−y, t)) ·y) [|y|2+ (f(x, t)−f(x−y, t))2]3/2dy +ρ3−ρ2 4πPV ZR2 ∇f(x, t)·y−∇g(x−y, t)·y [|y|2+ (f(x, t)−g(x−y, t))2]3/2dy. (6) In a similar way we obtain gt(x, t) =ρ3−ρ2 4πPV ZR2 (∇g(x, t)−∇g(x−y, t)) ·y) [|y|2+ (g(x, t)−g(x−y, t))2]3/2dy +ρ2−ρ1 4πPV ZR2 ∇g(x, t)·y−∇f(x−y, t)·y [|y|2+ (g(x, t)−f(x−y, t))2]3/2dy. (7) This coupled system describes the evolution of the moving boundaries Sf(x, t) and Sg(x, t). 3 Well-posedness in the stable scenario ρ1< ρ2< ρ3 Let us define the function d(f, g)(x, y, t) by the formula d(f, g)(x, y, t) = 1 [|y|2+ (f(x, t)−g(x−y, t))2]1/2∀x, y ∈R2, which measure the distance between the contours fand g. Therefore we consider f(x, t) approaching the value C∞as |x| → ∞ to avoid that the surfaces collapse at infinity. The section is devoted to prove the following theorem: 4 Theorem 3.1 Let f0(x)−C∞, g0(x)∈Hk(R2)for k≥4,d(f0, g0)∈L∞and ρ3> ρ2> ρ1. Then there exists a time T > 0so that there is a unique solution to (6) and (7) given by (f(x, t), g(x, t)) where f(x, t)−C∞, g(x, t)∈C1([0, T]; Hk(R2)),f(x, 0) = f0(x)and g(x, 0) = g0(x). Proof: We shall show that the following estimate holds d dtE(t)≤C(E(t) + 1)p(8) for universal constants (C,p) and the function E(t) defined by E(t) = kf−C∞kHk(t) + kgkHk(t) + kd(f, g)kL∞(t). Applying standard energy estimates argument permit us to conclude local existence. This is based on introducing a regularized version of the system (6) and (7) that allows to take limits satisfying uniformly the a priori bound (8). To simplify the exposition we shall consider ρ2−ρ1=ρ3−ρ2= 4πand k= 4, being the rest of the cases analogous. Some of the terms can be estimated exactly as in [7] and therefore we shall show below how to deal with the different ones. We have 1 2 d dtkf−C∞k2 L2(t) = ZR2 (f(x)−C∞)PV ZR2 (∇f(x)−∇f(x−y)) ·y [|y|2+ (f(x)−f(x−y))2]3/2dydx +ZR2 (f(x)−C∞)∇f(x)·P V ZR2 y [|y|2+ (f(x)−g(x−y))2]3/2 −ZR2 (f(x)−C∞)PV ZR2 ∇g(x−y)·y [|y|2+ (f(x)−g(x−y))2]3/2 =I1+I2+I3. For I1we decompose further: I1=J1+J2+J3where J1=Z|y|<1ZR2 (f(x)−C∞)(∇f(x)−∇f(x−y)) ·y [|y|2+ (f(x)−f(x−y))2]3/2dydx, J2=Z|y|>1 (f(x)−C∞)PV Z|y|>1 ∇f(x)·y [|y|2+ (f(x)−f(x−y))2]3/2dydx and J3=−Zy|>1 (f(x)−C∞)PV Z|y|>1 ∇f(x−y)·y [|y|2+ (f(x)−f(x−y))2]3/2dydx. Since ∂xif(x)−∂xif(x−y) = Z1 0∇∂xif(x+ (s−1)y)·y ds, 5 we get J1≤Z1 0 ds Z|y|<1|y|−1ZR2 |f(x)−C∞||∇2f(x+ (s−1)y)| [1 + ((f(x)−f(x−y))2|y|−2]3/2dxdy ≤Ckf−C∞kL2k∇fkH1. In the term J2integration by parts yields J2=3 2Z|y|>1ZR2|f(x)−C∞|2(f(x)−f(x−y))(∇f(x)−∇f(x−y)) ·y [|y|2+ ((f(x)−f(x−y))2]5/2dxdy ≤3 2Z|y|>1|y|−3ZR2|f(x)−C∞|2|f(x)−f(x−y)||y|−1(|∇f(x)|+|∇f(x−y)|) [1 + ((f(x)−f(x−y))2|y|−2]5/2dxdy ≤Ckf−C∞k2 L2k∇fk2 L∞≤Ckf−C∞k2 L2k∇fk2 H2. In a similar way for J3we have J3=Z|y|>1ZR2 (f(x)−C∞)(f(x−y)−C∞)2|y|2−(f(x)−f(x−y))2 [|y|2+ (f(x)−f(x−y))2]5/2dxdy + 3 Z|y|>1ZR2 (f(x)−C∞)(f(x−y)−C∞)(f(x)−f(x−y))∇f(x−y)·y [|y|2+ (f(x)−f(x−y))2]5/2dxdy −Z|y|=1 ZR2 (f(x)−C∞)(f(x−y)−C∞)|y|2 [|y|2+ (f(x)−f(x−y))2]3/2dxdσ(y) ≤C(k∇fkL∞+ 1)kf−C∞k2 L2. The term I2is estimated as follows I2=3 2ZR2ZR2 (f(x)−C∞)2(f(x)−g(x−y))(∇f(x)−∇g(x−y)) ·y [|y|2+ (f(x)−g(x−y))2]5/2dxdy = 3 Z|y|>1ZR2 dxdy + 3 Z|y|<1ZR2 dxdy I2≤3 2Z|y|>1|y|−3ZR2|f(x)−C∞|2(|∇f(x)|+|∇g(x−y)|)dxdy +3 2Z|y|<1ZR2|f(x)−C∞|2(|∇f(x)|+|∇g(x−y)|)(d(f, g)(x, y))3dxdy ≤Ckf−C∞k2 L2(k∇fkL∞+k∇gkL∞)(kd(f, g)k3 L∞+ 1) For I3we write I3=ZR2ZR2 (f(x)−C∞)g(x−y)2|y|2−(f(x)−g(x−y))2 [|y|2+ (f(x)−g(x−y))2]5/2dxdy +ZR2ZR2 (f(x)−C∞)g(x−y)3(f(x)−g(x−y))(∇f(x)−∇g(x−y)) ·y [|y|2+ (f(x)−g(x−y))2]5/2dxdy 6 and proceeding as before we find I3≤Ckf−C∞kL2kgkL2(k∇fkL∞+k∇gkL∞+ 1)(kd(f, g)k3 L∞+ 1). Thus Sobolev inequalities yields d dtkf−C∞k2 L2(t)≤C(E(t) + 1)p+1.(9) The analogous estimate for g is d dtkgk2 L2(t)≤C(E(t) + 1)p+1.(10) To estimate the higher order derivative, we consider the quantity 1 2 d dtk∂4 x1fk2 L2(t) = I4+I5+I6, where I4=ZR2 ∂4 x1f(x)∂4 x1PV ZR2 (∇f(x)−∇f(x−y)) ·y [|y|2+ (f(x)−f(x−y))2]3/2dydx, I5=ZR2 ∂4 x1f(x)∂4 x1PV ZR2 ∇f(x)·y [|y|2+ (f(x)−g(x−y))2]3/2dydx and I6=−ZR2 ∂4 x1f(x)∂4 x1PV ZR2 ∇g(x−y)·y [|y|2+ (f(x, t)−g(x−y, t))2]3/2dydx. The estimates for I4are obtained in [7]. We split I5and consider the most singular terms which are J4=ZR2 ∂4 x1f(x)∇∂4 x1f(x)·PV ZR2 y [|y|2+ (f(x)−g(x−y))2]3/2dydx, J5=−3ZR2 ∂4 x1f(x)ZR2 ∇f(x)·y(f(x)−g(x−y))(∂4 x1f(x)−∂4 x1g(x−y)) [|y|2+ (f(x)−g(x−y))2]5/2dydx, and J6=CZR2 ∂4 x1f(x)ZR2 ∇f(x)·y(f(x)−g(x−y))4(∂x1f(x)−∂x1g(x−y))4 [|y|2+ (f(x)−g(x−y))2]11/2dydx. For integral J4we find that J4=3 2ZR2ZR2|∂4 x1f(x)|2(f(x)−g(x−y))(∇f(x)−∇g(x−y)) ·y [|y|2+ (f(x)−g(x−y))2]5/2dxdy ≤Ck∂4 x1fk2 L2(k∇fkL∞+k∇gkL∞)(kd(f, g)k3 L∞+ 1). 7 For J5it follows J5=Z|y|>1|y|−3ZR2|∂4 x1f(x)||∇f(x)|(|∂4 x1f(x)|+|∂4 x1g(x−y)|)dxdy +Z|y|<1ZR2|∂4 x1f(x)||∇f(x)|(|∂4 x1f(x)|+|∂4 x1g(x−y)|)(d(f, g)(x, y))3dxdy ≤Ck∂4 x1fkL2k∇fkL∞(k∂4 x1fkL2+k∂4 x1gkL2)(kd(f, g)k3 L∞+ 1). Similarly we obtain J6=CZ|y|>1|y|−6ZR2|∂4 x1f(x)||∇f(x)|(|∂x1f(x)|+|∂x1g(x−y)|)4dxdy +CZ|y|<1ZR2|∂4 x1f(x)||∇f(x)|(|∂x1f(x)|+|∂x1g(x−y)|)4(d(f, g)(x, y))6dxdy ≤Ck∂4 x1fkL2k∇fkL2(k∂x1fkL∞+k∂x1gkL∞)4(kd(f, g)k6 L∞+ 1) and finally d dtk∂4 x1fk2 L2(t)≤C(E(t) + 1)p+1. Analogously we get d dtk∂4 x2fk2 L2(t)≤C(E(t) + 1)p+1, and therefore using (9) it follows d dtkf−C∞k2 H4(t)≤C(E(t) + 1)p+1.(11) In order to estimate the H4norm of gwe proceed as for the surface fto obtain d dtkgk2 H4(t)≤C(E(t) + 1)p+1.(12) Next, we analyze the evolution of the distance between the surfaces. For the quantity d(f, g)(x, y) we have d dtd(f, g)(x, y, t) = −(f(x)−g(x−y, t))(ft(x)−gt(x−y)) [|y|2+ (f(x)−g(x−y))2]3/2 ≤(d(f, g)(x, y, t))2(kftkL∞(t) + kgtkL∞(t)). (13) We now estimate kftkL∞(t), being equivalent to control kgtkL∞(t). We consider for ft(x, t) the following splitting: I7=PV ZR2 (∇f(x, t)−∇f(x−y, t)) ·y [|y|2+ (f(x, t)−f(x−y, t))2]3/2dy, 8 I8=PV ZR2 ∇f(x, t)·y [|y|2+ (f(x, t)−g(x−y, t))2]3/2dy, I9=−PV ZR2 ∇g(x−y, t)·y [|y|2+ (f(x, t)−g(x−y, t))2]3/2dy. Let us decompose further: I7=J7+J8+J9where J7=Z|y|<1 (∇f(x, t)−∇f(x−y, t)) ·y [|y|2+ (f(x, t)−f(x−y, t))2]3/2dy, J8=∇f(x, t)·PV Z|y|>1 y [|y|2+ (f(x, t)−f(x−y, t))2]3/2dy and J9=−PV Z|y|>1 ∇f(x−y, t)·y [|y|2+ (f(x, t)−f(x−y, t))2]3/2dy. Hence |J7| ≤ CZ|y|<1|y|−1dyk∇2fkL∞≤Ck∇2fkL∞. We rewrite J8as J8=∇f(x, t)·Z|y|>1 y |y|3 1−[1 + (f(x, t)−f(x−y, t))2|y|−2]3/2 [1 + (f(x, t)−f(x−y, t))2|y|−2]3/2dy, and considering the function Q(α) = [1 + α2]3/2, the mean value theorem gives J8=∇f(x, t)·Z|y|>1 y |y|4−3[1 + β2]1/2β|f(x)−f(x−y)| [1 + (f(x, t)−f(x−y, t))2|y|−2]3/2dy, where 0 ≤β≤ |f(x)−f(x−y)||y|−1. Therefore |J8| ≤ Ck∇fkL∞kf−C∞kL∞. In a similar manner as for J3, integrations by parts in J9yields |J9| ≤ C(k∇fkL∞+ 1)kf−C∞kL∞. For I8we split I8=PV Z|y|>1 dy +Z|y|<1 dy =K1+K2. We deal with K1as we have done with J8to get |K1| ≤ k∇fkL∞(kf−C∞kL∞+C∞+kgkL∞). 9 [18] M. Siegel, R. Caflisch and S. Howison. Global Existence, Singular Solutions, and Ill- Posedness for the Muskat Problem. Comm. Pure and Appl. Math., 57: 1374-1411, 2004. [19] E. Stein. Harmonic Analysis. Princeton University Press. Princeton, NJ, 1993. 16